Time complexity of two disjoint simple paths

Finding two disjoint simple paths on two given sets of points is a geometric problem introduced by je erickson. this problem has various applications in computational geometry, e.g. robot motion planning, generating polygon, etc. we will present a reduction from planar hamiltonian path to this problem, and prove that it is npcomplete. to the best of our knowledge, no study has considered its complexity up until now. we also present a reduction from planar hamiltonian path problem to the problem of nding a path on given points in the presence of arbitrary obstacles and prove that it is also np-complete. also, we present a heuristic algorithm with time complexity of o(n^4) to solve this problem. the proposed algorithm rst calculates the convex hull for each of the entry points and then produces two simple paths on the two entry point sets.